As the deadline for Brexit approaches, the British newspapers clarify their positions on what they consider to be the best course of action111https://bit.ly/2txNgpn. The three possibilities currently under consideration are (1) accepting Theresa May’s deal with the EU (D), (2) leaving the EU with no deal (N), and (3) postponing Brexit or canceling it altogether (R). Among the major newspapers, the most pro-Leave position is taken by The Daily Telegraph, which is strongly critical of May’s deal, and ranks it below no deal; it is also strongly opposed to any delays, so its ranking can be described as . On the opposite end of the spectrum is The Guardian, which backs remaining in the EU, but views May’s deal as (scarcely) more acceptable than no deal at all, so its ranking is . The Times and The Sun take a more moderate position: both back May’s deal, but The Times views no deal as the most catastrophic option, while The Sun is firmly opposed to any delays to Brexit. Thus, if we order the newspapers according to their political stance, from left to right, the rankings change from to to to .
These preferences have the property that for any two of the available alternatives , if the first newspaper in our order ranks over then all newspapers that rank over appear before all newspapers that rank over , i.e., every pair of alternatives ‘crosses’ at most once. An ordered list of rankings (an election) that has this property is known as single-crossing (see Section 2 for formal definitions). Single-crossing elections have many attractive properties and have recently received a lot of attention in computational social choice literature; see, e.g., the survey by ELP-trends ELP-trends. Besides the example in the previous paragraph, there are real-life settings where we expect to observe essentially single-crossing preferences. For instance, when a country is about to introduce a flat tax rate and the voters are asked to rank several options (say, 25%, 28%, 33%, 45%), they have to consider the tradeoff between the amount they will have to pay and the value of the government services that can be provided at a given level of taxation; if all voters apply this reasoning, the election where the voters are ordered by income should be single-crossing (this example dates back to mir:j:single-crossing mir:j:single-crossing.
However, the examples we described, while single-crossing in spirit, may not be single-crossing in a formal sense, because the single-crossing property is very fragile. For instance, in the Brexit scenario, if we expand the list of newspapers to include a broader sample of publications, we may find left-leaning newspapers that support no deal. In the tax scenario, not all voters may be capable of evaluating the consequences of each choice.
Now, we can easily check whether an election where voters are ordered according to a publicly observable parameter is single-crossing, simply by looking at all pairs of candidates. However, as argued above, we expect the answer to be ‘no’. A more important—and more challenging!—task is to understand whether this election is close to single-crossing, i.e., can be made single-crossing by deleting a few voters or candidates or swapping a few pairs of candidates, as this will tell us whether the observable parameter used to order the voters is relevant for understanding the voters’ preferences. This knowledge is important, e.g., for managing electoral campaigns, as it can be used to identify the segments of voting population that are more likely to support a given candidate (which can help the campaign managers to decide whom to target in a get-out-the-vote effort).
We introduce a mapping between elections and graphs that enables us to use powerful graph-theoretic machinery to analyze nearly single-crossing elections. Briefly, given an election over a set of candidates (i.e., an ordered list of linear orders over ), we build an undirected graph that has as its set of vertices, and contains an edge between two candidates and if and only if and cross more than once in ; we say that implements . In other words, the graph documents obstacles that prevent from being single-crossing; in particular, an independent set in corresponds to a subset of candidates such that the restriction of to this subset is single-crossing. We then ask which graphs are -implementable, i.e., can be implemented by elections with voters. We show (Section 5) that we can obtain any undirected graph in this manner; in fact, the number of voters required is bounded by a linear function in the size of the graph. However, for any constant there are graphs that are not -implementable. In Section 4 we focus on -implementable graphs and obtain a complete characterization of this class of graphs by relating it to the class of permutation graphs. We also argue that all -implementable graphs are comparability graphs; importantly, every comparability graph is a perfect graph.
Our results have implications for the problem of deciding whether an election is nearly single-crossing with respect to the given order of voters (Section 6). In particular, we use our mapping to establish the hardness of computing two measures of how close a given election is to being single-crossing: one of these measures is based on deleting as few candidates as possible, and the other is based on splitting the candidates into as few groups as possible. On the other hand, our results for -implementable graphs enable us to show that the problems we consider are in P for elections with voters.
The concept of single-crossing elections has been proposed in the social choice literature several decades ago [Mirrlees1971, Roberts1977]. Single-crossing elections are appealing both from a purely social choice-theoretic perspective and from a computational perspective: for instance, their weak majority relation is necessarily transitive [Mirrlees1971] and they admit efficient algorithms for determining a winning committee under a well-known committee selection rule whose output is hard to compute for general preferences [Skowron et al.2015]. Several groups of authors have considered the problem of identifying nearly single-crossing elections [Bredereck et al.2016, Cornaz et al.2013, Elkind and Lackner2014, Jaeckle et al.2018], but in all these papers the authors assumed that there was no publicly observable parameter that determined the ordering of the voters, i.e., they considered the problem of reordering the voters so that the resulting election can be made single-crossing by applying a small number of modifications; in contrast, we assume that the order of voters is fixed.
Our analysis is similar in spirit to the research on implementation of directed graphs as majority graphs. In this line of work, the input is a directed graph with a vertex set , and the goal is to construct an election over the set of candidates such that there is a directed edge from to in the input graph if and only if a strict majority of voters in prefer to . The classic McGarvey theorem [McGarvey1953] establishes that every directed graph can be implemented in this way using at most two voters per edge, and subsequent work has reduced this number to [Stearns1959, Erdos and Moser1964]; Corollary 9 in Section 5 can be viewed as an analogue of McGarvey’s theorem in our setting. Recently, brandt-few brandt-few investigated what directed graphs can be implemented by elections with two or three voters; this research is similar to our analysis in Section 4.
For each , we denote the set by .
Elections An election is a pair , where is a finite set of candidates and is a list of votes. Each vote , , is a linear order over . We refer to elements of as voters; thus, is the vote of voter . We will sometimes use the term ‘profile’ to refer to , and we use the terms ‘vote’, ‘preference’ and ‘ranking’ interchangeably. We say that voter prefers to or ranks over (and write ) if precedes in the linear order . A restriction of an election with to a subset of candidates is an election , where and for every pair of candidates and every it holds that is ranked above in if and only if is ranked above in .
Single-crossing (also known as intermediate or order-restricted) preferences capture settings where the voters can be ordered along a single axis according to their preferences.
An election with is single-crossing if for every pair of candidates with there is a such that .
We emphasize that we define single-crossing elections with respect to a fixed order of the voters, i.e., we are interested in settings where voters are ordered according to a publicly observable parameter.
Graphs An undirected graph is a pair , where is a finite set, and is a collection of size- subsets of . The elements of are called vertices, and the elements of are called edges. For readability, we will sometimes write instead of . We assume that the reader is familiar with the definitions of a path, a cycle, a tree, and a bipartite graph. A hole is a cycle with such that if and only if or . An anti-hole is a sequence of distinct vertices with such that if and only if or .
A directed graph is a pair , where is a finite set, and
is a collection of ordered pairs of elements of. An element of is called an arc; an arc points from vertex to vertex . An undirected graph can be turned into a directed graph by choosing an orientation for each edge , i.e., transforming into or . A directed graph is said to be transitive if for every triple of vertices such that and it holds that .
3 Single-Crossing Implementation
A pair of candidates is a multi-crossing pair in an election with if , , and for some with .
The multi-crossing graph of an election is an undirected graph such that and if and only if is a multi-crossing pair in . An election implements an undirected graph if . We say that a graph is -implementable if there exists an -voter election that implements it. Since the set of candidates in an election that implements a graph is necessarily , we often omit from the notation and speak of a profile that implements .
By definition, the only graph that is -implementable is the graph with no edges. Thus, in the remainder of the paper we study graphs that are -implementable for .
To build the reader’s intuition, we first consider -implementation. We show how to implement several families of graphs, such as paths, trees and even-length cycles. While for some of these families their -implementability follows from the more general results in Section 4.2, the proofs below provide efficient algorithms for finding a -voter profile that implements a given graph. Also, we relate -implementable graphs to other well-known classes of graphs, such as permutation graphs and comparability graphs. Finally, we prove that some graphs are not -implementable.
First, it is easy to see that we can -implement empty graphs and cliques: an empty graph is implemented by a profile where all three voters are identical, and a clique can be implemented by a profile where the first and the third voter rank the candidates in the same order, and the second voter ranks the candidates in the opposite order. A somewhat more complex construction establishes that all paths and even-length cycles are -implementable.
There is a polynomial-time algorithm that given a graph that is a path or an even-length cycle constructs a -voter profile that implements .
Suppose that is a path. For convenience, we assume that and . To start, we construct a -voter profile where all voters rank the candidates as . Then, we modify the preferences of the first and the third voter by swapping candidates and in her rankings, for . Also, we modify the preferences of the second voter by swapping candidates and in her ranking, for . Table 1 (left) illustrates the resulting profiles for . To see why this profile implements a path, consider an even-numbered candidate , . By construction, all voters rank above all candidates with and below all candidates with . On the other hand, voters 1 and 3 rank below and above , whereas voter 2 ranks above and below . Thus, is a multi-crossing pair if and only if
. For odd-numbered candidateswith , as well as for candidates and , the argument is similar.
A similar approach can be used if is an even-length cycle; we omit the proof, but provide an example in Table 1(right). ∎
The reader may wonder why we only consider cycles of even length in Proposition 1. Now, the cycle of length is -implementable because it is a clique. However, for the cycle of length is not -implementable; this follows from Theorem 5 in Section 4.2.
On the other hand, we can extend the result of Proposition 1 to arbitrary trees.
There is a polynomial-time algorithm that given a graph that is a tree constructs a -voter profile that implements .
Given a tree , we pick an arbitrary vertex to be its root. Our implementation is recursive. We observe that an isolated vertex is trivially implementable. Then we consider a vertex of whose children are , . We show that, if for each we have a -implementation of the tree rooted at in which the first voter ranks first, then we can construct a -implementation of the tree rooted at in which the first voter ranks first. Using this idea, we can construct an implementation of starting from the leaves and ending at .
Fix a vertex that is not a leaf. Let be the children of , and for each let be the subtree of rooted at ; let be the set of vertices of . Suppose that for each we have a -implementation of in which is ranked first in the first vote.
We will first stack these three implementations on top of each other: we construct a -voter profile where each voter ranks all candidates in above all candidates in for all , and for each and each the -th voter ranks above if and only if the -th voter in the given -implementation of ranks above . Then we pull the candidates to the top of the first vote: we modify the preferences of voter so that she ranks in position for and the relative order of the other candidates remains unchanged. Note that this step does not introduce any multi-crossing pairs.
In remains to insert into the voters’ rankings. To this end, in the first vote we insert after the first candidates, in the second vote we place on top, and in the third vote we place last. Note that for each the pair is multi-crossing, but for every and every the pair is not multi-crossing, as both of the first two voters rank above . Thus, we have implemented the edges connecting to its children. However, in the resulting profile voter does not rank first. To remedy this, we first reverse the order of candidates in each vote and then reverse the order of votes; neither of these operations changes the set of multi-crossing pairs, and in the resulting profile is ranked first. Each of these steps can be implemented in polynomial time. ∎
4.2 General Constructions
We can relate -implementable graphs to two well-known classes of graphs: permutation graphs and comparability graphs.
Definition 3 (Permutation graph).
An undirected graph is a permutation graph if there exist permutations of such that if and only if appears before in exactly one of the permutations and .
It can be decided in polynomial time whether a given graph is a permutation graph; moreover, if the answer is ‘yes’, the respective permutations can be constructed in polynomial time as well [Golumbic1980, Simon and Trunz1994].
It is immediate that every permutation graph can be implemented by a -voter profile.
Every permutation graph is -implementable, and a -voter profile that implements it can be computed in polynomial time.
Let be a permutation graph, and let and be two permutations that witness this. Then the profile implements . ∎
In fact, the proof of Theorem 3 suggests a stronger claim: an undirected graph is a permutation graph if and only if it can be implemented by a -voter profile where the first and the third voter have the same preferences. Recall that our implementation of cliques has this property, but our implementation of even-length cycles does not. There is a reason for this: it is not hard to show that cycles of length at least are not permutation graphs. Thus, permutation graphs form a proper subclass of -implementable graphs. The following proposition further clarifies the relationship between -implementable graphs and permutation graphs.
A graph is -implementable if and only if there exist two permutation graphs and such that .
Let and be two permutation graphs on the same set of vertices . Let (respectively, ) be a pair of permutations witnessing that (respectively, ) is a permutation graph. Note that if a pair of permutations witnesses that a given graph is a permutation graph, then so does the pair of permutations , for any given permutation . Applying this observation to with , we can assume that . Then the profile where for each voter ranks the candidates according to implements : a pair of candidates is multi-crossing in this profile if and only if both and disagree with on the order of and .
Conversely, let be a -implementation of a graph . Consider the graphs and implemented, respectively, by and . As argued in the proof of Theorem 3, both of these graphs are permutation graphs, and a pair of candidates is multi-crossing in if and only if and . This completes the proof. ∎
Another relevant class of graphs is comparability graphs.
Definition 4 (Comparability graph).
A graph is a comparability graph if edges in can be oriented so that the resulting directed graph is transitive.
Every -implementable graph is a comparability graph.
Consider a graph implemented by a profile . We orient the edge from to if and from to otherwise. Since is a multi-crossing pair, implies . Consider a pair of arcs , in the resulting directed graph. We have , and hence . Similarly, , implies and , implies . Thus, our directed graph also contains the arc . ∎
Comparability graphs are known to be perfect graphs [Mirsky1971], i.e., graphs that contain neither odd-length holes nor odd-length anti-holes222Originally, perfect graphs are defined as graphs with the property that the chromatic number of every induced subgraph is equal to the size of the maximum clique in that subgraph [Berge1961]; however, by the strong Berge conjecture, which was proved by perfect perfect, perfect graphs are exactly the graphs with no odd-length holes and no odd-length antiholes.. Hence, Theorem 5 explains why Proposition 1 does not extend to odd cycles: by definition, odd cycles are not perfect graphs. Also, it subsumes the existence results of Propositions 1 and 2: paths, even-length cycles, and trees can be easily seen to be comparability graphs (we note, however, that these propositions also provide efficient algorithms to compute the respective -voter profiles, and it is not clear how to extract such algorithms from the proof of Theorem 5). In particular, every bipartite graph is a comparability graph (we can direct the edges from one part to the other), and paths, even-length cycles and trees are bipartite graphs. However, there exists a bipartite graph that is not -implementable (and hence -implementable graphs form a proper subclass of comparability graphs).
The bipartite -regular graph with parts of size each (see Figure 1) is not -implementable.
5 -Implementation for
We have seen that not all graphs are -implementable. However, we will now show that every graph is implementable by an election whose number of voters is linear in . We first define a class of single-crossing elections that can be used to implement an arbitrary graph.
A single-crossing election with is fully single-crossing if for every pair of candidates with there is an such that , , and voter ranks just above .
Note that in a fully single-crossing election the ranking of the last voter is the inverse of the ranking of the first voter, i.e., every pair of candidates ‘crosses’ exactly once.
If there exists a fully single-crossing election with , then every -vertex graph is -implementable.
Consider a fully single-crossing election with , , and let be an -vertex graph. Let . By construction, the election is single-crossing. Now, for each edge we identify an such that in we have , , and voter ranks just above . We then swap and in the preferences of the -st voter in (who, like voter in the original election, ranks just above prior to the swap). This ensures that is a multi-crossing pair in the resulting election. In the end we obtain an election that implements . ∎
A fully single-crossing election with candidates and voters can be obtained as a maximal chain in a weak Bruhat order; in this election, which we will denote by , each vote differs from its predecessor by exactly one swap of adjacent candidates (see [Bredereck et al.2013] and the references therein). One can use as a starting point to implement an arbitrary graph with votes: we take , remove each vote that is obtained from its predecessor by swapping a pair of candidates that does not correspond to an edge in the input graph, and then use the construction in the proof of Theorem 7. However, for dense graphs this produces an implementation with voters.
In contrast, our next theorem, in conjunction with Theorem 7, shows that every graph is -implementable. Our construction is inspired by the concept of odd-even sort [Lakshmivarahan et al.1984] and, to the best of our knowledge, is new.
For every there exists a fully single-crossing election with candidates and voters.
Let . Consider the following sequence of votes. The first vote is given by . Then, for each , the vote is obtained from the vote by swapping the candidates in positions and for . Similarly, for each , the vote is obtained from the vote by swapping candidates in positions and for . The resulting profile for is given in Table 2. We will now argue that this procedure produces a fully single-crossing profile.
First, we show that in the candidates are ranked as . Indeed, suppose that is even. Then, by construction, for , in the -th vote is ranked in position . Then, in vote candidate remains ranked in position , and in the remaining votes moves down step by step, ending up in position . Conversely, if is odd, it moves down step by step in the first votes, stays in the last position for one more step, and then starts climbing back up, ending in position . This implies our claim.
We have shown that each pair of candidates is swapped at least once. To see that it is swapped exactly once, we compute the total number of swaps. If is even, then every even-numbered vote differs from its predecessor by swaps and every odd-numbered vote apart from the first vote differs from its predecessor by swaps. Thus, the total number of swaps is , and hence each pair of candidates is swapped exactly once. For odd values of the calculation is similar.
It remains to note that, by construction, if , but , then is ranked just above in , as we only swap adjacent candidates. This completes the proof. ∎
An undirected graph is -implementable.
The bound in Corollary 9 is linear in . One can ask if we implement each graph using a constant number of votes. It turns out that the answer is ‘no’.
To show this, we use the Erdös–Szekeres theorem [Erdös and Szekeres1935] to argue that if a graph is implementable by an election with a few voters then it has to have a large clique or a large independent set.
If an -vertex graph is -implementable then it has a clique or size at least or an independent set of size at least .
On the other hand, we have the following well-known fact, which can be easily proved by the probabilistic method (see, e.g., [Bollobás and Erdös1976]).
There exists an integer constant such that for every positive integer there exists a graph with vertices with the property that each clique and each independent set in have at most vertices.
For every positive integer there exists a graph with that is not -implementable.
We will now apply the tools developed in Sections 4 and 5 to the problem of detecting elections that are close to being single-crossing with respect to a given order of voters, for two measures of closeness.
An instance of Candidate Deletion is given by an election and an integer . It is a yes-instance if and only if there is a subset with such that is single-crossing. An instance of -Candidate Partition is given by an election . It is a yes-instance if and only if can be partitioned into sets so that for each the election is single-crossing.
We will now show that both of these problems are hard, by leveraging our observation that is single-crossing if and only if is an independent set in .
Candidate Deletion is NP-complete; -Candidate Partition is NP-complete for every .
It is immediate that both of these problems are in NP. To show that Candidate Deletion is NP-hard, we reduce from Independent Set. An instance of Independent Set is given by a graph and an integer ; it is a yes-instance if has an independent set of size at least and a no-instance otherwise. This problem is well-known to be NP-hard [Garey and Johnson1979]. Given an instance of the Independent Set problem, we build an election that implements it using the construction described in Section 5; the size of the resulting election is polynomial in the size of , and has an independent set of size at least if and only if is a yes-instance of Candidate Deletion.
We use the same argument for -Candidate Partition; the only difference is that we reduce from the -Coloring problem. An instance of -Coloring is given by a graph ; it is a yes-instance if there exists a mapping such that for every and a no-instance otherwise. Note that each ‘color’ , , forms an independent set in . The -Coloring problem is well-known to be NP-hard for every [Garey and Johnson1979]. Again, given a graph , we construct an election that implements it, and observe that is -colorable if and only if is a yes-instance of -Candidate Partition. ∎
On the other hand, we can use the results in Section 4 to show that Candidate Deletion and -Candidate Partition are in P for elections with at most voters.
Given an election with at most three voters and an integer , we can decide in polynomial time whether the pair is a yes-instance of Candidate Deletion. Also, for each we can decide in polynomial time whether is a yes-instance of -Candidate Partition.
Given an election with at most three voters, we construct its multi-crossing graph . By Theorem 5 the graph is a comparability graph and hence a perfect graph. As argued in the proof of Theorem 13, to decide whether is a yes-instance of Candidate Deletion, it suffices to determine whether is a yes-instance of Independent Set, and to decide whether is a yes-instance of -Candidate Partition, it suffices to determine whether is a yes-instance of -Coloring. It remains to note that both Independent Set and -Coloring are known to be polynomial-time solvable on perfect graphs (see, e.g., diestel diestel). ∎
By a similar argument, -Candidate Partition is polynomial-time solvable for any number of voters.
-Candidate Partition is polynomial-time solvable.
An election is a yes-instance of -Candidate Partition if and only if the graph is -colorable, and -colorability can be checked in polynomial time. ∎
We have introduced the notion of single-crossing implementation of a graph and showed how to exploit the connection between elections and graphs to better understand the complexity of detecting elections that are nearly single-crossing with respect to a fixed order of voters. Our approach turned out to be useful for two distance measures: the number of candidates that need to be deleted to make the input election single-crossing, and the number of parts that the candidate set needs to be split into so that the projection of the input election onto each set is single-crossing. There are other distance measures that can be used in this context: e.g., we can remove or partition voters, or swap adjacent candidates in voters’ preferences. In a companion paper [Lakhani et al.2019], we explore the complexity of computing how far a given election is from being single-crossing according to several other distance measures.
Our work suggests several interesting open questions. First, it is not known what is the smallest value of such that every -vertex graph is -implementable: there is a significant gap between the upper bound of Corollary 9 and the lower bound of Theorem 12. Second, our characterization of -implementable graphs does not suggest an efficient algorithm for checking whether a graph is -implementable. More broadly, we do not know if one can efficiently compute the smallest profile that implements a given graph; we conjecture that this problem is NP-complete.
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